Radiation wave function

In 1913 Niels Bohr proposed a system of rules that defined a specific set of discrete orbits for the electrons of an atom with a given atomic number. These rules required the electrons to exist only in these orbits, so that they did not radiate energy continuously as in classical electromagnetism. This model was extended first by Sommerfeld and then by Goudsmit and Uhlenbeck. In 1925 Heisenberg, and in 1926 Schrn dinger, proposed a matrix or wave mechanics theory that has developed into quantum mechanics, in which all of these properties are included. In this theory the state of the electron is described by a wave function from which the electron s properties can be deduced. 

In Equation 12.11, v is the frequency of incident radiation (cm-1), v o is the frequency corresponding to the energy difference between ground and excited electronic state, the sum is over all excited states, and d0n and d o are the dipole transition moments between the ground and excited state (dno = (n d 0) = / I nd I o dx, the T s are wave functions and d is the dipole moment operator). At low frequency (v -> 0) Equation 12.11 reduces to the static field expression... 

The interaction of a system with the electromagnetic radiation leads to a time-dependence of the wave function ip, which follows from the time-dependent Schrodinger equation... 

As pointed out by Dexter (41), the over-all theoretical problems of luminescence are exceedingly difficult to treat even in their simplest forms. This is true not only because they involve simultaneous interactions among radiation, matter, and phonons, but also because the specific details of the wave functions are of first-order importance. 

Let us consider two stationary states n and m of an unperturbed system represented by the wave function V and such that Em > Let us assume that at / = 0, the system is in the state n. At this time, the system comes under the perturbing influence of the radiation of a range of frequencies in the neighbourhood of vm of a definite field strength E. 

The symmetries of the initial and the final wave functions and of the electromagnetic radiation operator determine the allowedness or forbiddenness of an electronic transition. The transition moment integrand must be totally symmetric for an allowed transition such that Mmn V0. 

We now complete the discussion of Section XII-C by taking into account the independent decay of the initial state, 5>, due to interaction with the radiation field. Let Ftot(t)> denote the wave function which describes the state of our system at time t, in the case where both the perturbations F and HiDt are considered. Then for the overall probability of finding the molecule in the initial state, S>, at time t > 0, we have... 

There exists another more consistent way of obtaining the electron transition operators. We can start with the quantum-electrodynamical description of the interaction of the electromagnetic field with an atom. In this case we find the relativistic operators of electronic transitions with respect to the relativistic wave functions. After that they may be transformed to the well-known non-relativistic ones, accounting for the relativistic effects, if necessary, as corrections to the usual non-relativistic operators. Here we shall consider the latter in more detail. It gives us a closed system of universal expressions for the operators of electronic transitions, suitable to describe practically the radiation in any atom or ion, including very highly ionized atoms as well as the transitions of any multipolarity and any type of radiation (electric or magnetic). 

Let us consider the non-relativistic limit of the relativistic operators describing radiation. Expressing the small components of the four-component wave functions (bispinors) in terms of the large ones and expanding the spherical Bessel functions in a power series in cor/c, we obtain, in the non-relativistic limit, the following two alternative expressions for the probability of electric multipole radiation ... 

Thus, the kind and quantity of relativistic corrections to the length and velocity forms of 1-radiation are different. From this point of view the concept of the equivalency of these forms must be improved both forms will lead to coinciding transition values for the accurate (exact) wave functions only if we account for the relativistic corrections of order v2/c2 to the transition operators (in practice, only for the velocity form). The other conclusion accounting for the relativistic effects leads to qualitatively new results, namely, to new operators, which allow not only improved values of permitted transitions, but also describe a number of lines, which earlier were forbidden. These relativistic corrections usually are very small, but they are very important for weak intercombination lines of light neutral atoms (see Chapter 30). 

A submatrix element of the Ek-radiation with regard to antisymmetric wave functions for the relativistic analogue of transition (25.2) is as... 

As was shown in Chapter 4 (formula (4.22)), relativistic corrections of the order a2 to the intercombination 1-transitions in length form for accurate wave functions compensate each other. It follows from formulas (4.18)-(4.20) that for the velocity form of the 1-transition operator the relativistic corrections are of the order a2 and may be presented in length, velocity and acceleration forms. Calculations of the 1-radiation for the Be isoelectronic sequence (Z = 4 92) indicate that these relativistic cor-... 

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